Mathematical Modelling and Control

: Compared to many infectious diseases, tuberculosis has a high mortality rate. Because of this, a great deal of illustrative research has been done on the modeling and study of tuberculosis using mathematics. In this work, a mathematical model is created by taking into account the underlying presumptions of this disease. One of the main novelties of the paper is to consider two di ﬀ erent treatment strategies namely protective treatment for the latent populations from the disease and the main treatment applied to the infected populations. This situation can be regarded as the other novelty of the paper. The susceptible, latent, infected, and recovered populations, as well as the two mentioned treatment classes, are all included in the proposed six-dimensional model’s compartmental framework. Additionally, a region that is biologically possible is presented, as well as the solution’s positivity, existence, and uniqueness. The suggested model’s solutions are carried out as numerical simulations using assumed and literature-based parameter values and analyzing its graphics. To get the results, a fourth-order Runge-Kutta numerical approach is used.


Introduction
Humanity has fought epidemics for centuries, and millions of people have lost their lives. From time to time, these outbreaks manifest themselves in the form of plague, smallpox, HIV/AIDS, SARS, avian flu, and influenza. One of these epidemics is tuberculosis (TB) which is popularly known. Tuberculosis is a long-term bacterial and infectious disease caused by the microbe "mycobacterium tuberculosis (MTB)". The way of transmission is often caused by sputum that a tuberculosis patient spits into the environment, or by Bacillus-laden droplets that are scattered when coughing. Although tuberculosis is an infection that can occur in more than one organ, it is mainly observed in the lungs and mediastinal lymph nodes of the lungs. Bacillus is emitted from an active tuberculosis patient by coughing, sneezing, or other means through droplets into the air in saliva, and the infection is spread by removing particles suspended in the air. It is one of the oldest known diseases, that can be maintained, although still continues to be one of the world's most common and deadly infectious diseases, and more than three million people per year are died due to TB [1].
People who experience tuberculosis can continue their healthy lives for months without any symptoms. During this period, the person's immune system tries to prevent the development of the disease by fighting against the MTB bacterium. But in cases where the immune system cannot show sufficient resistance, tuberculosis microbes become active and tuberculosis disease occurs [2]. (1994) [5] presented a mathematical model to study the accelerating effect of HIV infection on TB disease, while Castillo-Chavez and Feng (1997) [6] revealed differences between two TB individuals with and without drug-resistant TB. In another study, it was stated that the effect of exogen on the qualitative dynamics of TB was too great [7].
Monte-Carlo simulations were conducted to determine the likelihood that 10,000 clinical patients receiving different doses of moxifloxacin could reach or exceed the point of exposure to the drug required to suppress their resistance to moxifloxacin in TB [8]. In addition, it was concluded that advanced TB diagnostic techniques have a significant impact on t-related disease and death rates in HIV endemic areas, and it was emphasized that as TB rates continue to increase, advanced diagnostic techniques should be considered as TB control strategies [9].
Analysis of the mathematical model created in research on multidrug-resistant (CID) and common drug-resistant (YID) strains in South Africa, the region with the highest TB rate worldwide, has yielded important results for the next 10 years. It showed that the spread of TB culture and drug sensitivity among adults in South Africa could save more than 47.000 lives and prevent more than 7.000 cases of CID-TB in the period from 2008 to 2017. This corresponds to a 17% decrease in total TB deaths and a 47% decrease in CID-TB deaths [10]. Bowong and Tewa (2009) [11] proved that the TB system they studied was asymptotically globally stable and has a single stable equilibrium, and showed that depending on the basic reproductive rate, this stable structure occurs either in a regional disease state or in have also been conducted in relation to significant area of science and infectious illnesses including COVID-19 [21][22][23][24][25][26][27][28][29][30][31][32], cancer cells-cancer stem cells [33], optimal control and bifurcation [34], alcoholism [35], cholera [36,37], Parkinson's disease [38], disturbance effect in intracellular calcium dynamic on fibroblast cells [39], HIV [40,41], babesiosis [42], predator-prey [43,44], viral system with the non-cytolytic immune assumption [45], Nipah virus [46], an epidemic model with general interference function and high-order perturbation [47], potential scenarios for wastewater treatment [48], Gompertz growth model [49], a general epidemic model with logistic growth [50], synchronization [51], a sewage treatment model [52].

Model formulation
In the present TB model, population compartments consist of six populations: Susceptible (S ), Latent (L), Treatment for latently infected individuals (T 1 ),   The model is therefore given by the following set of ordinary differential equations: subject to the initial conditions (IC), where (S (t), L(t), T 1 (t), I(t), T 2 (t), R(t)) ∈ R 6 + . The functions S (t), L(t), T 1 (t), I(t), T 2 (t), R(t) and their derivatives are considered to be continuous at t ≥ 0 in this situation.

Analysis of the model
The positivity and boundedness of the solution for the recommended model (2.1) are given in this section. The existence conditions and stability findings for the equilibria are then given.

Positivity of solutions and determining the biologically invariant region
We begin with the following theorem to this subsection: Proof 1. According to the study's suggestion [55], we evaluate the first equation while accounting for the nonlinear system of equations (2.1): By integrating Equation (3.2) and using the exponential growth condition, we get this gives Theorem 2. In area A ⊂ R 6 + , given by following, the solutions of system (2.1) with IC (2.2) are specified: (3.5) Proof 2. By taking the total population, we have (3.6) L(0) Initial Latent population 830 [54] T1(0) Initial Treatment 1 population 0 [54] I(0) Initial Infected population 801 [54] T2(0) Initial Treatment 2 population 0 assumed R(0) Initial Recovered population 0 [54] Then we have the following for the whole population Equation (3.7)'s solution is presented as 8) where N 0 = N(0) is the definition of the beginning population. With the aid of the Birkhoff-Rota theorem, we can state that if N 0 < Λ µ , then as t → ∞, asymptotically N(t) → Λ µ in Eq (3.5), and the overall population size becomes N(t) → Λ µ , then 0 ≤ N ≤ Λ µ . As a result, region A is where all of the model's viable solutions converge [57].
Since the kernels satisfy the Lipschitz condition (see Then, using the final inequality, we arrive at: We derive the following theorem from these findings.  I (t), T 2 (t) and R (t) are bounded and their kernels ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 and ϕ 6 hold the Lipschitz condition, we can give the following by taking Eq (3.18) into account, The functions in Eq (3.20) are now demonstrated to be the solutions of the specified TB model. We believe (3.21) Then we now present that the terms stated in Eq (3.21) and n (t) ≤ t 0 ϕ 6 (τ, R) − ϕ 6 (τ, R n−1 ) dτ ≤ t 0 ϕ 6 (τ, R) − ϕ 6 (τ, R n−1 ) dτ recursively carrying out this procedure, we obtain and n (t) ≤ {t} n+1 q n 6 Υ.
Considering these last two inequalities at t max point, we have The final step is taken, after applying the limit on both sides of the final inequalities as n → ∞, and by taking into account the theorem's 3 conclusions, we get ω Theorem 5. The TB model constructed in the paper has a unique solution.

Proof 5. Suppose that there is a different systemic solution,
such as S 1 (t), L 1 (t), T 1 1 (t), I 1 (t), T 2 1 (t) and R 1 (t). After that we get When both sides of Eq (3.23) are subjected to the norm, we get As a result of the kernels ϕ 1 , ϕ 2 , ϕ 3 , ϕ 4 , ϕ 5 and ϕ 6 satisfying the Lipschitz criterion, we can write

Mathematical Modelling and Control
Volume 3, Issue 2, 88-103 which gives As a result, the model is shown to have a unique solution, proving the theorem.

Equilibria, stability and reproduction number
To determine the system (3.7) equilibrium points, we set: Six steady states are produced by solving system Eq (4.1) collectively. According to their biological importance, we provide these equilibria and describe their local behavior.
The disease-free equilibrium (DFE), which is denoted as the first equilibrium point, stated as Ω 1 = ( Λ µ , 0, 0, 0, 0, 0), this indicates that there is no cell population. Co-equilibrium point, which is the second equilibrium point, is given as The third equilibrium is another co-equilibrium point presented by Finally, the endemic equilibrium is given as In the next subsection, we proceed with the evaluation of the reproduction number so that one can have some idea about the dynamics of the disease by using this value.
Moreover, we show the local stability results of equilibria that have been obtained for the system in the form of theorems and proofs.

Basic reproduction number
The population group that the model we are considering here assumes is heterogeneous, with non-homogeneous individuals that have been grouped. First, we provide the solution set as: then we use the next generation matrix approach described in [58] to evaluate the system (2.1)'s fundamental reproduction number. Following that, we define the solution set provided by Eq (4.2) as the difference between two matrices: The matrix generation approach is used to define F = ∂F i (Ω 1 ) ∂t j and V = ∂V i (Ω 1 ) ∂t j , 1 ≤ i, j ≤ 2 for the F and V matrices of uninfected division at the equilibrium point Ω 1 . As more explicitly, they can be written in the following forms: where the matrix V is non-singular and the matrix F is non-negative. With the use of the matrix FV −1 's spectral radius at the equilibrium point Ω 1 , the disease's fundamental reproduction number is computed, which is indicated by two cases namely R 0 1 and R 0 2 : Theorem 6. In the epidemic model, the disease-free equilibrium point Ω 1 is locally asymptotically stable (LAS) if R 0 < 1, else unstable.
Proof 6. Now, we take into consideration the following Jakobian matrix to highlight the stability criteria at the DFE point indicated as Ω 1 .
The characteristic equation (CE) of matrix J(Ω 1 ) can thus be found as: are the solutions of the CE. It is obvious that λ 1,2,4,6 are negative. Furthermore, we can obtain that λ 3 is negative if R 0 1 < 1 and λ 5 is negative as well if R 0 2 < 1, which means that if R 0 = max R 0 1 , R 0 2 < 1 the DFE is LAS. The proof is finished with this.
Theorem 7. The second equilibrium point Ω 2 of the epidemic model is locally asymptotically stable if R 0 1 > 1, otherwise there is at least one unbounded solution.
Proof 7. In order for the stability of Ω 2 (λ), we have the following characteristic equation: where λ 1 = −µ, λ 2 = −(m + µ + σ), λ 3 = −(c + µ + σ), The remaining two roots, namely λ 5 and λ 6 , satisfy the following equation: . According to the Routh-Hurwitz stability criteria second order [59,60], the coefficients of the quadratic characteristic equation given in Eq (4.4) must satisfy the following conditions if the roots of the equation have negative real part: (4.5) which means that the following inequalities hold:

Numerical results
The significance of numerical results for the TB model we developed by taking the consciousness parameter into account is covered in this section. In the present paper, for the simulation, we have taken the parameter values as given in Table 1. In Table 1, we have considered the values as a parameter of "year". We have utilized the parameter value as the year in all of our simulations and graphs. We have demonstrated the efficiency of the parameter β in 4, which represents the rate of susceptible individuals to enter the hidden TB compartment, on the latent and infected classes. By considering Figure 4, it can be concluded that as the parameter increases from 0.0008 to 0.0023, when the populations in L increase, the population of I decreases.
In Figure 5, we have rededicated the consciousness effect for the number of infected individuals with TB. According  to Figure 5, we have concluded that as the parameter f increases from 0.6 to 0.85, the number of infected individuals with TB decreases. This result is very important in terms of determining a parameter that helps to reduce the number of TB cases.
In Figure 6, in relation to the parameter z, we have shown the population density of the R class. The figure makes it evident that as the parameter's values rise from 0.4 to 0.9, the density of the R class rises along with it.

Concluding remarks
In this study, we have constructed an S LT 1 IT 2 R model that contains an effective consciousness strategy for the TB epidemic disease. With the help of the is very important in terms of determining a parameter that helps to reduce the number of TB cases. We display the population density of the recovered people in relation to parameter z in Figure 6, which is the proportion of infected individuals recovering without treatment.
The impacts of the treatment and being educated thoughts can be highlighted in future studies by employing appropriate control strategies. Additionally, the fractional order can be taken into consideration when applying the integer-order model.